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Given a group $G$ acting cocompactly on a smooth manifold $M$ by decktransformations, there is an integration map, defined recently by Kato,Kishimoto and Tsutaya, from the top-degree bounded de Rham cohomology of $M$ tothe coinvariants $\ell^\infty(G)_G$. We generalize its definition and show thatit is an isomorphism. In the presence of boundary, a relative version ofbounded de Rham cohomology is considered.

Bounded differential forms and coinvariants of bounded functions